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Journal of Convex Analysis 21 (2014), No. 1, 189--200
Copyright Heldermann Verlag 2014

Some Geometric Properties of the CesÓro Function Spaces

Damian Kubiak
Mathematics Department, Tennessee Technological University, 110 University Drive, Box 5054, Cookeville, TN 38505, U.S.A.


Some geometric properties of the Ces{\`a}ro function spaces $C_{p,w}$, $1\leqslant p<\infty$, induced by an arbitrary positive weight function $w$ on an interval $(0,l)$ where $0 < l \leqslant\infty$ are studied in this paper. It is shown that all non-empty relatively weakly open sets in the unit ball of $C_{p,w}$ have diameter $2$. Also $C_{p,w}$, $1<p<\infty$ is strictly convex but no point of its unit ball is strongly extreme. Moreover, some connections between uniformly non-square points and various geometric properties in general Banach spaces are presented.

Keywords: Cesaro function space, diameter 2 property, weak neighborhoods, uniformly non-square points.

MSC: 46E30, 46B20, 46B42

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